On the One-Leg Methods for Solving Nonlinear Neutral Differential Equations with Variable Delay

Based on A-stable one-leg methods and linear interpolations, we introduce four algorithms for solving neutral differential equations with variable delay. A natural question is which algorithm is better. To answer this question, we analyse the error behavior of the four algorithms and obtain their error bounds under a one-sided Lipschitz condition and some classical Lipschitz conditions. After extensively numerically experimenting, we give a positive conclusion.

[1]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[2]  Wayne H. Enright,et al.  A delay differential equation solver based on a continuous Runge–Kutta method with defect control , 1997, Numerical Algorithms.

[3]  Z. Jackiewicz.  Adams methods for neutral functional differential equations , 1982 .

[4]  Nicola Guglielmi,et al.  Numerical stability of nonlinear delay differential equations of neutral type , 2000 .

[5]  Z. Jackiewicz.  Quasilinear multistep methods and variable step predictor-corrector methods for neutral functional differential equations , 1986 .

[6]  Christopher T. H. Baker,et al.  Discontinuous solutions of neutral delay differential equations , 2006 .

[7]  Wan-Sheng Wang,et al.  On the one-leg theta-methods for solving nonlinear neutral functional differential equations , 2007, Appl. Math. Comput..

[8]  Chen,et al.  D-CONVERGENCE OF ONE-LEG METHODS FOR STIFF DELAY DIFFERENTIAL EQUATIONS , 2001 .

[9]  Christopher T. H. Baker,et al.  The tracking of derivative discontinuities in systems of delay-differential equations , 1992 .

[10]  Z. Jackiewicz.  Variable-step variable-order algorithm for the numerical solution of neutral functional differential equations , 1987 .

[11]  Lin,et al.  THE STABILITY OF LINEAR MULTISTEP METHODS FOR LINEAR SYSTEMS OF NEUTRAL DIFFERENTIAL EQUATIONS , 2001 .

[12]  Wan-Sheng Wang,et al.  Convergence of waveform relaxation methods for neutral delay differential equations , 2008, Math. Comput. Model..

[13]  Shoufu Li,et al.  Stability Analysis of Theta-Methods for Nonlinear Neutral Functional Differential Equations , 2008, SIAM J. Sci. Comput..

[14]  Wansheng Wang,et al.  Nonlinear stability of general linear methods for neutral delay differential equations , 2008 .

[15]  K. J. in 't Hout.  A note on unconditional maximum norm contractivity of diagonally split Runge-Kutta methods , 1996 .

[16]  Zdzislaw Jackiewicz,et al.  ONE-STEP METHODS OF ANY ORDER FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS. , 1984 .

[17]  Zhifeng Liu,et al.  A Copula-Based Correlation Measure and its Application in Chinese Stock Market , 2009, Int. J. Inf. Technol. Decis. Mak..

[18]  Chengming Huang,et al.  Stability and error analysis of one-leg methods for nonlinear delay differential equations , 1999 .

[19]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[20]  Christopher T. H. Baker,et al.  Retarded differential equations , 2000 .

[21]  W. Enright,et al.  Convergence Analysis of the Solution of Retarded and Neutral Delay Differential Equations by Continuous Numerical Methods , 1998 .

[22]  Wan-Sheng Wang,et al.  Nonlinear stability of Runge-Kutta methods for neutral delay differential equations , 2008 .

[23]  Zdzislaw Jackiewicz,et al.  Numerical solution of neutral functional differential equations by Adams methods in divided difference form , 2006 .

[24]  Hongjiong Tian,et al.  The asymptotic stability of one-parameter methods for neutral differential equations , 1994 .

[25]  Wansheng Wang,et al.  Nonlinear stability of theta-methods for neutral differential equations in Banach space , 2008, Appl. Math. Comput..

[26]  L. Torelli,et al.  Stability of numerical methods for delay differential equations , 1989 .

[27]  M. Zennaro.  Natural continuous extensions of Runge-Kutta methods , 1986 .

[28]  Xiaoguang Yang,et al.  Skewness of return distribution and coefficient of risk premium , 2009, J. Syst. Sci. Complex..

[29]  R. N. Castleton,et al.  A First Order Method for Differential Equations of Neutral Type , 1973 .

[30]  Guang-Da Hu,et al.  Stability analysis of numerical methods for systems of neutral delay-differential equations , 1995 .

[31]  Z. Bartoszewski,et al.  Delay dependent estimates for waveform relaxation methods for neutral differential-functional systems , 2004 .

[32]  Zhang,et al.  NONLINEAR STABILITY OF NATURAL RUNGE-KUTTA METHODS FOR NEUTRAL DELAY DIFFERENTIAL EQUATIONS , 2002 .

[33]  Kolmanovskii,et al.  Introduction to the Theory and Applications of Functional Differential Equations , 1999 .

[34]  Rossana Vermiglio,et al.  A Stable Numerical Approach for Implicit Non-Linear Neutral Delay Differential Equations , 2003 .

[35]  Yunkang Liu.  Numerical Solution of Implicit Neutral Functional Differential Equations , 1999 .

[36]  Zdzislaw Jackiewicz,et al.  The numerical solution of neutral functional differential equations by Adams predictor—corrector methods , 1991 .

[37]  Nonlinear stability of one-leg methods for delay differential equations of neutral type , 2008 .

[38]  M. Zennaro.  Natural continuous extensions of Runge-Kutta formulas , 1986 .

[39]  Nicola Guglielmi,et al.  Open issues in devising software for the numerical solution of implicit delay differential equations , 2006 .

[40]  A. Bellen,et al.  Numerical methods for delay differential equations , 2003 .

[41]  C. Paul.  Designing efficient software for solving delay differential equations , 2000 .

[42]  Fathalla A. Rihan,et al.  Numerical modelling in biosciences using delay differential equations , 2000 .

[43]  Germund Dahlquist,et al.  G-stability is equivalent toA-stability , 1978 .

[44]  Lin Qiu,et al.  The NGP-stability of Runge-Kutta methods for systems of neutral delay differential equations , 1999, Numerische Mathematik.

[45]  Josef Stoer,et al.  Numerische Mathematik 1 , 1989 .

[46]  Zdzislaw Jackiewicz,et al.  Waveform Relaxation Methods for Functional Differential Systems of Neutral Type , 1997 .

[47]  田红炯,et al.  The Asymptotic Stability of Neutral Equations , 1994 .